\documentclass{article}
  \usepackage{amsmath}
  \usepackage{amssymb}
  \newcommand{\setcomp}[1] {{#1}^{\mathsf{c}}}

\begin{document}

\begin{enumerate}
  \item Prove $B\setminus A = B\cap \setcomp{A}$.
  \item Prove the following are equivalent:
  $A\subseteq B, A \cap B = A, A\cup B = B$.

  \item Prove the Distributive Law:
  $A\cap(B\cap C) = (A\cap B)\cup (A\cap C)$.

  \item Write the dual of:
  $
  (\textbf{U} \cap A)\cup (B\cap A) = A,
  (A\cap \textbf{U})\cap (\emptyset \cup \setcomp{A}) = \emptyset
  $.

  \item Prove $(A\cup B)\setminus (A\cap B) = (A\setminus B)\cup (B\setminus A)$.

  \item Prove:

    a. $(A\cap B)\cup (A\cap \setcomp{B}) = A$

    b. $A\cup B = (A\cap \setcomp{B})\cup (\setcomp{A}\cap B)\cup (A\cap B)$

  \item Prove $n(P(S)) = 2^{n(S)}$ if $S$ is a finite set.

  \item Try to figure out the formula for: $n(partition(S))$.

  \item Let $[A_1, A_2, \cdots, A_m]$ and $[B_1, B_2, \cdots, B_m]$ be partitions of a set S.
  Prove the following collection is also a partition (called the \textsl{cross partition}) of S:
  $$
  P = [A_i\cap B_j | i = 1, \cdots, m, j = 1, \cdots, n]\setminus \emptyset
  $$

  \item Prove the following properties of the symmetric difference:

    a. Associative Law $(A\oplus B)\oplus C = A\oplus(B \oplus C)$

    b. Comutative Law $A\oplus B = B\oplus A$

    c. Cancellation Law $A\oplus B = A\oplus C \implies B = C$

    d. Distributive Law $A\cap (B\oplus C) = (A\cap B)\oplus (A\cap C)$

  \item Consider $m$ nonempty distinct sets, $A_1, A_2, \cdots, A_m$ in a universal set $\textbf{U}$. Prove:

    a. There are $2^m$ fundamental products of the $m$ sets.

    b. Any two fundamental products are disjoint.

    c. $\textbf{U}$ is the union of all the fundamental products.

\end{enumerate}

\end{document}
